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arXiv:cond-mat/0212602 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Full reduction of large finite random Ising systems by real space renormalization group

Avishay Efrat, Moshe Schwartz

Published Aug 18, 2003• 3 citations
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Abstract

We describe how to evaluate approximately various physical interesting quantities in random Ising systems by direct renormalization of a finite system. The renormalization procedure is used to reduce the number of degrees of freedom to a number that is small enough, enabling direct summing over the surviving spins. This procedure can be used to obtain averages of functions of the surviving spins. We show how to evaluate averages that involve spins that do not survive the renormalization procedure. We show, for the random field Ising model, how to obtain $\overline{\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{r})}=\overline{〈\ensuremath{\sigma}(0)\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{r})〉\ensuremath{-}〈\ensuremath{\sigma}(0)〉〈\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{r})〉},$ the ``connected'' correlation function, and $\overline{S(\stackrel{\ensuremath{\rightarrow}}{r})}=\overline{〈\ensuremath{\sigma}(0)\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{r})〉},$ the ``disconnected'' correlation function. Consequently, we show how to obtain the average susceptibility and the average energy. For an Ising system with random bonds and random fields, we show how to obtain the average specific heat. We conclude by presenting our numerical results for the average susceptibility and the function $\overline{\ensuremath{\Gamma}(r)}$ along one of the principal axes. (In this work, the full three-dimensional (3D) correlation is calculated and not just parameters such $\ensuremath{\nu}$ or $\ensuremath{\eta}).$ The results for the average susceptibility are used to extract the critical temperature and critical exponents of the 3D random field Ising system.

DOI
10.1103/physreve.68.026114

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arXiv:cond-mat/9507044•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jan 1, 1996•81 citations
We describe how to evaluate approximately various physical interesting quantities in random Ising systems by direct renormalization of a finite system. The renormalization procedure is used to reduce the number of degrees of freedom to a number that is small enough, enabling direct summing over the surviving spins. This procedure can be used to obtain averages of functions of the surviving spins. We show how to evaluate averages that involve spins that do not survive the renormalization procedure. We show, for the random field Ising model, how to obtain \ensuremathΓ(r\ensuremath→)‾=〈\ensuremathσ(0)\ensuremathσ(r\ensuremath→)〉\ensuremath−〈\ensuremathσ(0)〉〈\ensuremathσ(r\ensuremath→)〉‾,\overline{\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{r})}=\overline{〈\ensuremath{\sigma}(0)\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{r})〉\ensuremath{-}〈\ensuremath{\sigma}(0)〉〈\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{r})〉},\ensuremathΓ(r\ensuremath→)​=〈\ensuremathσ(0)\ensuremathσ(r\ensuremath→)〉\ensuremath−〈\ensuremathσ(0)〉〈\ensuremathσ(r\ensuremath→)〉​, the connected'' correlation function, and $\overline{S(\stackrel{\ensuremath{\rightarrow}}{r})}=\overline{〈\ensuremath{\sigma}(0)\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{r})〉},$ the disconnected'' correlation function. Consequently, we show how to obtain the average susceptibility and the average energy. For an Ising system with random bonds and random fields, we show how to obtain the average specific heat. We conclude by presenting our numerical results for the average susceptibility and the function \ensuremathΓ(r)‾\overline{\ensuremath{\Gamma}(r)}\ensuremathΓ(r)​ along one of the principal axes. (In this work, the full three-dimensional (3D) correlation is calculated and not just parameters such \ensuremathν\ensuremath{\nu}\ensuremathν or \ensuremathη).\ensuremath{\eta}).\ensuremathη). The results for the average susceptibility are used to extract the critical temperature and critical exponents of the 3D random field Ising system.
It was recently shown [Phys. Rev. Lett. 110, 227201 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.227201] that the critical behavior of the random-field Ising model in three dimensions is ruled by a single universality class. This conclusion was reached only after a proper taming of the large scaling corrections of the model by applying a combined approach of various techniques, coming from the zero- and positive-temperature toolboxes of statistical physics. In the present contribution we provide a detailed description of this combined scheme, explaining in detail the zero-temperature numerical scheme and developing the generalized fluctuation-dissipation formula that allowed us to compute connected and disconnected correlation functions of the model. We discuss the error evolution of our method and we illustrate the infinite limit-size extrapolation of several observables within phenomenological renormalization. We present an extension of the quotients method that allows us to obtain estimates of the critical exponent α of the specific heat of the model via the scaling of the bond energy and we discuss the self-averaging properties of the system and the algorithmic aspects of the maximum-flow algorithm used.
The random-field Ising model is one of the few disordered systems where the perturbative renormalization group can be carried out to all orders of perturbation theory. This analysis predicts dimensional reduction, i.e., that the critical properties of the random-field Ising model in D dimensions are identical to those of the pure Ising ferromagnet in D-2 dimensions. It is well known that dimensional reduction is not true in three dimensions, thus invalidating the perturbative renormalization group prediction. Here, we report high-precision numerical simulations of the 5D random-field Ising model at zero temperature. We illustrate universality by comparing different probability distributions for the random fields. We compute all the relevant critical exponents (including the critical slowing down exponent for the ground-state finding algorithm), as well as several other renormalization-group invariants. The estimated values of the critical exponents of the 5D random-field Ising model are statistically compatible to those of the pure 3D Ising ferromagnet. These results support the restoration of dimensional reduction at D=5. We thus conclude that the failure of the perturbative renormalization group is a low-dimensional phenomenon. We close our contribution by comparing universal quantities for the random-field problem at dimensions 3≤D<6 to their values in the pure Ising model at D-2 dimensions, and we provide a clear verification of the Rushbrooke equality at all studied dimensions.
The authors discuss results from two types of real space renormalization group (RSRG) calculations applied to the random field Ising model in three dimensions. Starting from a lattice of size L, the RSRG is used to reduce the lattice to a size L=2, on which the trace is done exactly. In this way, thermodynamic properties, such as the magnetization and susceptibility, can be determined approximately. They find that, for a given size, the susceptibility increases as the temperature, T, is reduced down to the transition temperature, T c , and becomes essentially independent of temperature below T c . Both in the vicinity of T c and at lower temperatures, there are large sample-to-sample fluctuations in the susceptibility which grow with increasing system size. They interpret these results in terms of the droplet theory of the transition.
Phase transition in d=3d=3d=3 ferromagnetic Ising models with random fields is analyzed directly at the zero-temperature critical point. Critical behavior is extracted from correlation functions averaged over an ensemble of exact ground states obtained with a new integer optimization algorithm. For Gaussian distribution of random fields finite-size scaling demonstrates a continuous phase transition with effective disconnected susceptibility exponent $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}\ensuremath{\approx}\ensuremath{-}0.9$, correlation-length exponent $\ensuremath{\nu}\ensuremath{\approx}1.0$, and magnetization exponent $\ensuremath{\beta}\ensuremath{\approx}0.05$.
We investigate the critical properties of the d=3d=3d=3 random-field Ising model with a Gaussian field distribution at zero temperature. By implementing suitable graph-theoretical algorithms, we perform a large-scale numerical simulation of the model for a vast range of values of the disorder strength hhh and system sizes $\mathcal{V}=L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$, with $L\ensuremath{\le}156$. Using the sample-to-sample fluctuations of various quantities and proper finite-size scaling techniques we estimate with high accuracy the critical disorder strength hc{h}_{\mathrm{c}}hc​ and the correlation length exponent $\ensuremath{\nu}$. Additional simulations in the area of the estimated critical-field strength and relevant scaling analysis of the bond energy suggest bounds for the specific heat critical exponent $\ensuremath{\alpha}$ and the violation of the hyperscaling exponent $\ensuremath{\theta}$. Finally, a data collapse analysis of the order parameter and disconnected susceptibility provides accurate estimates for the critical exponent ratios $\ensuremath{\beta}/\ensuremath{\nu}$ and $\overline{\ensuremath{\gamma}}/\ensuremath{\nu}$, respectively.
Using a cluster-flipping Monte Carlo algorithm combined with a generalization of the histogram reweighting scheme of Ferrenberg and Swendsen [Phys. Rev. Lett. 61, 2635 (1988); 63, 1195 (1989)], we have studied the equilibrium properties of the thermal random-field Ising model on a cubic lattice in three dimensions. We have equilibrated systems of L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L spins, with values of L up to 32, and for these systems the cluster-flipping method appears to a large extent to overcome the slow equilibration seen in single-spin-flip methods. From the results of our simulations we have extracted values for the critical exponents and the critical temperature and randomness of the model by finite size scaling. For the exponents we find \ensuremath{\nu}=1.02\ifmmode\pm\else\textpm\fi{}0.06, \ensuremath{\beta}=0.06\ifmmode\pm\else\textpm\fi{}0.07, \ensuremath{\gamma}=1.9\ifmmode\pm\else\textpm\fi{}0.2, and \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2.9\ifmmode\pm\else\textpm\fi{}0.2, where \ensuremath{\nu}, \ensuremath{\beta}, \ensuremath{\gamma}, and \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{} govern the critical singularities in the correlation length, magnetization, and connected and disconnected susceptibilities, respectively. \textcopyright{} 1996 The American Physical Society.